The Dirac Delta Function

11 Key excerpts on "The Dirac Delta Function"

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  eBook - ePub

  Theory of Differential Equations in Engineering and Mechanics

  • Kam Tim Chau(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  8.10 DIRAC DELTA FUNCTION

  For simplicity, we first recast the definition of a Dirac delta function defined in (8.8 ) and (8.9 ) for the one-dimensional case:

  (8.161)

  δ ( x ) = 0 x ≠ 0

  = ∞ x = 0

  (8.162)

  ∫

  - ∞

  ∞

  δ

  ( x )

  d x = 1

  This can be considered as a special case of the one given in (8.8 ) and (8.9 ). Another basic definition of the delta function is the shifting property of The Dirac Delta Function:

  (8.163)

  ∫

  - ∞

  ∞

  f

  ( x )

  δ

  ( x )

  d x = f

  ( 0 )

  This Dirac delta function is named in honor of electrical engineer and physicist, Paul Dirac, who received the Nobel Prize in physics for his major contribution to quantum mechanics. The idea of using The Dirac Delta Function was actually much earlier than its usage in quantum mechanics by Dirac. For example, it can be used to prescribe point force in the case of beam bending subject to a concentrated force. Physically, in the domain ofmechanics, the delta function defined above is a point force or called a concentrated point.

  According to our normal understanding in mathematics, there is no function that is nonzero only at one point and yet still has a finite integral as defined in (8.162

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  Digital Signal Processing

  • (Author)
  • 2014(Publication Date)
  • Research World

    (Publisher)

  An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy. Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corresponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac Delta Function as such was introduced as a convenient notation by Paul Dirac in his influential 1927 book Principles of Quantum Mechanics . He called it the delta function since he used it as a continuous analogue of the discrete Kronecker delta. Definitions The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, and which is also constrained to satisfy the identity ____________________ WORLD TECHNOLOGIES ____________________ This is merely a heuristic definition. The Dirac delta is not a true function, as no function has the above properties. Moreover there exist descriptions of the delta function which differ from the above conceptualization. For example, sinc( x / a )/ a (where sinc is the sinc function) becomes the delta function in the limit as a → 0, yet this function does not approach zero for values of x outside the origin, rather it oscillates between 1/ x and −1/ x more and more rapidly as a approaches zero. The Dirac Delta Function can be rigorously defined either as a distribution or as a measure. As a measure One way to rigorously define the delta function is as a measure, which accepts as an argument a subset A of the real line R , and returns δ ( A ) = 1 if 0 ∈ A , and δ ( A ) = 0 otherwise.

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  Generalized Functions

  • (Author)
  • 2014(Publication Date)
  • White Word Publications

    (Publisher)

  In applied mathematics, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero. An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy. Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corresponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac Delta Function as such was introduced as a convenient notation by Paul Dirac in his influential 1927 book Principles of Quantum Mechanics . He called it the delta function since he used it as a continuous analogue of the discrete Kronecker delta. Definitions The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, and which is also constrained to satisfy the identity ________________________ WORLD TECHNOLOGIES ________________________ This is merely a heuristic definition. The Dirac delta is not a true function, as no function has the above properties. Moreover there exist descriptions of the delta function which differ from the above conceptualization. For example, sinc( x / a )/ a (where sinc is the sinc function) becomes the delta function in the limit as a → 0, yet this function does not approach zero for values of x outside the origin, rather it oscillates between 1/ x and −1/ x more and more rapidly as a approaches zero.

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  Mathematical Analysis & Generalized Functions

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  In applied mathematics, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero. An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy. Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corres-ponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac Delta Function as such was introduced as a convenient notation by Paul Dirac in his influential 1927 book Principles of Quantum Mechanics . He called it the delta function since he used it as a continuous analogue of the discrete Kronecker delta. Definitions The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite, and which is also constrained to satisfy the identity ________________________ WORLD TECHNOLOGIES ________________________ This is merely a heuristic definition. The Dirac delta is not a true function, as no function has the above properties. Moreover there exist descriptions of the delta function which differ from the above conceptualization. For example, sinc( x / a )/ a (where sinc is the sinc function) becomes the delta function in the limit as a → 0, yet this function does not approach zero for values of x outside the origin, rather it oscillates between 1/ x and −1/ x more and more rapidly as a approaches zero.

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  Quantum Mechanics I

  A Problem Text

  • David DeBruyne, Larry Sorensen(Authors)
  • 2018(Publication Date)
  • Sciendo

    (Publisher)

  Dirac indicates it is “ ... merely a convenient notation, enabling us to express in a concise form certain relations which we could, if necessary, rewrite in a form not involving improper functions, but only in a cumbersome way which would tend to obscure the argument 1 .” If the system is discrete, a Kronecker delta is appropriate to model an “all or none” quantity. If the system is continuous, a Dirac delta function is appropriate to model an “all or none” situation. Likely the most straightforward graph of a delta function is zero everywhere except for a tall, thin rectangle inside a domain of length Δ x . The area of the rectangle is length × width, or Δ x 1 Δ x = 1 If the domain is diminished, the function must become “taller” to preserve the area at 1. In the limit of Δ x → 0 , the functional value goes to ∞ , but the area is still 1. The delta function is an idealization of an infinitely high, infinitely thin spike with an area of 1. The mathematical interpretation of The Dirac Delta Function is varied. Some view it as an actual function, others view it as a generalized function 2 , others classify it as a distribution. Some consider it nothing more than a pathological mathematical object, devoid of recognition. Dirac considered it an improper function. The delta function is an extension of the Kronecker delta to continuous systems. All continuous systems involve infinities which present problems that do not exist in finite systems. All calculus problems involve infinities. That calculus is useful is evidence that the sum (or difference or quotient or...) of some infinities converge. The problem of convergence remains a research topic within multiple realms of mathematics. We do not address the 1 Dirac, The Principles of Quantum Mechanics (Clarendon Press, Oxford, England, 1958), 4th ed., pp. 58-59. 2 Lighthill, An Introduction to Fourier Analysis and Generalised Functions (Cambridge Univer-sity Press, Cambridge, England, 1958). 118

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  Advanced Structural Dynamics

  • Eduardo Kausel(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  619 619 9 Mathematical Tools 9.1 Dirac Delta and Related Singularity Functions Dirac delta functions are extremely useful in dynamics, because they allow to represent in a compact mathematical fashion loads or masses that are concentrated at a point in space (i.e., a point load or a lumped mass), or that are impulsive in time (e.g., a hammer blow). A Dirac delta function δ(x − a) can be visualized as a rectangular function, or window, centered at x = a, which has small width w and large height 1/ w, as shown in Figure 9.1 . Its area is then w(1/ w) = 1, that is, it is unity. In the limit when w goes to zero, the func- tion becomes infinitely large, while its width becomes infinitesimally small, but its area remains unity. When this box function is used as a weighting function in an integral involving an arbitrary function f(x) over an interval containing a (i.e., x 1 < a < x 2 ), it is easy to see that f x x a dx Lim f x w dx f a w w f a x x w a w a w ( ) ( ) ( ) ( ) ( ) / / δ − = = = ∫ ∫ → − + 1 2 0 2 2 1 1 (9.1) Thus, an integral with The Dirac Delta Function simply reproduces the integrand at the location of the singularity. Functions such as the one above, which are defined in terms of the limit to an integral, are commonly referred to as singularity functions or distributions. It should be noticed that there are many ways of approaching this Dirac delta function. For example, instead of the box function, we could also have used a triangular function of width 2w and height 1/ w that is centered at x = a. Its area is again unity. Many other representations are also possible. By using the Dirac delta singularity function, we can now concisely model a concen- trated load P acting at x = a in terms of a distributed load: b x P x a ( ) ( ) = − δ (9.2) An impulsive, concentrated load occurring at time T and acting at x = a would then be of the form b x t P x a t T ( , ) ( ) ( ) = − − δ δ (9.3)

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  eBook - ePub

  Delta Functions

  Introduction to Generalised Functions

  • R F Hoskins(Author)
  • 2009(Publication Date)
  • Woodhead Publishing

    (Publisher)

  Principles of Quantum Mechanics (Third ed., Oxford, 1947). Arguably what he says there ought to be sufficient for any reader with adequate knowledge of classical analysis to understand the significance of the notation which Dirac introduces, and to make successful and trouble-free use of it.

  But, as many students have found to their cost, the conceptual basis on which the delta function is supposed to be founded is far from clear, and a more rigorous basic approach is essential to prevent misunderstanding. The most popular, and perhaps the most easily grasped, theory of the delta function is to treat it as, in a certain sense, the limit of a sequence of ordinary functions. What is more, this kind of approach can be readily extended to allow the definition of a wide range of other generalised functions, over and above the delta function itself, and its derivatives. It is given a straightforward and comprehensive treatment in the justly celebrated book, An Introduction to Fourier Analysis and Generalised Functions (C.U.P. 1960) by Sir James Lighthill . This is strongly recommended as supplementary reading, although it may be found somewhat more demanding than the treatment given here.

  A mine of general information on the theory, applications and history of the delta function is to be found in the classic text Operational Calculus based on the two-sided Laplace integral by Balth. Van der Pol and H.Bremmer (C.U.P. 1955). Although published over 50 years ago this is still a most valuable reference text and is again strongly recommended for supplementary reading. However, the reader should perhaps be warned that this book does, unfortunately, introduce yet another variant notation for the Heaviside unit step function; this appears as U (t ) and, more importantly, is constrained to assume the value

  1 2

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  Partial Differential Equations

  A First Course

  • Rustum Choksi(Author)
  • 2022(Publication Date)
  • American Mathematical Society

    (Publisher)

  In other words, take ? ′ (?) = { 0, ? ≠ 0, +∞, ? = 0. (5.6) Unfortunately there is ambiguity in this +∞ ; for example, in order to reconstruct ? (up to a constant) from its derivative we would need to make sense of the recurring product “ 0 × +∞ ”. Further to this point, suppose we considered a larger jump, say ? 7 (?) = { 0, ? < 0, 7, ? ≥ 0, 3 In fact even one of the greatest mathematicians / computer scientists of the 20th century, John von Neumann (1903–1957) , was so adamant that Dirac’s delta function was “ mathematical fiction ” that he wrote his monumental mono-graph, published in 1932, Mathematische Grundlagen der Quantenmechanik (Mathematical Foundations of Quantum Me-chanics) in part to explain quantum mechanics with absolutely no mention of the delta “function”. John von Neumann described this artifact in the preface as an “ improper function with self-contradictory properties ”. This counterpart to Dirac’s treatise was based on functional analysis and Hilbert spaces. 4 Named after the British self-taught scientist and mathematician Oliver Heaviside (1850–1925) , a rather interesting person who was quite critical about contemporary mathematical education. Look up online his “Letter to Nature”. 158 5. The Delta “Function” and Distributions in One Space Dimension would its derivative still be (5.6)? The functions ?(?) and ? 7 (?) are clearly different functions whose structure is lost in the derivative definition (5.6). Let us pursue this further by focusing on integration . Fix a smooth ( ? 1 ) function 𝜙 which is identically 0 if |?| ≥ 1 . Consider the function ?(?) (which, as we will later see, is called a convolution) defined by ?(?) ≔ ∫ ∞ −∞ ?(? − ?) 𝜙(?) ??, where ? is the Heaviside function of (5.5). While the definition of ?(?) is rather “con-voluted”, it presents a perfectly well-defined function. Indeed, despite the discontinuity in ? , ? is continuous and, in fact, differentiable at all ? .

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  Green's Functions and Linear Differential Equations

  Theory, Applications, and Computation

  • Prem K. Kythe(Author)
  • 2011(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  2.1. GENERALIZED FUNCTIONS 23 the δ -function has a nonzero dimension. For example, the dimension of the δ -function of time δ ( t ) is equal to the inverse time, i.e., the dimension of frequency because, by definition, ∞ −∞ δ ( t ) dt = 1 , which is dimensionless. In other words, the dimension of δ ( t ) is equal to the dimension of the inverse function 1 /t . The Dirac δ -function is used in electronics to represent the unit impulse and sampled signals. It also represents a unit mass ideally concentrated at the origin; if concentrated elsewhere, e.g., at the integer n , it is denoted by δ n , and the δ -function translated at n is defined by δ n ( t ) = δ ( t − n ) . Although the above definition of the δ -function is heuristically relevant, it is not mathematically correct. In the Riemann sense the integral of the δ -function defined by (2.10) is not well-defined, or in the Lebesgue sense it equals zero. However, for each ε > 0 the integral T ε [ φ ] = ∞ −∞ f ε ( t ) φ ( t ) dt (2.13) exists for any fixed continuous function φ , and it converges to the value φ (0) as ε → 0 + ; that is, lim ε → 0 + T ε [ φ ] = T [ φ ] = φ (0) . This definition can provide us with a valid definition of δ -function as a limit of the integrals of type (2.13). In fact, the integral (2.13) defines a linear functional on the test function φ ( t ) , generated by the function f ε ( t ) which is known as the kernel of the functional T ε . The notion of a functional is more general than that of a function because a functional depends on a variable which is a function itself but its values are real numbers. The linear functional T ε [ φ ] on the test function φ is generated by the integral (2.13) with the kernel f ε ( t ) , where the test function φ belongs to the class D of smooth functions in C ∞ 0 ( R ) with compact support. The functional that assigns to each test function its value at 0 will correspond to the Dirac delta distribution.

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  Fourier Methods in Imaging

  • Roger L. Easton Jr.(Author)
  • 2010(Publication Date)
  • Wiley

    (Publisher)

  A value may be assigned by using the Cauchy principal value, which is the mean value of the integral as x → 0 from the negative and positive sides of the origin. The resulting expression is identical to the definition of STEP[x] in Equation (6.10):  x −∞ δ[α] dα =    1 if x > 0 1 2 if x = 0 0 if x < 0    = STEP[x] (6.102) By differentiating both sides of this result and applying the fundamental theorem of calculus, we obtain another useful expression for The Dirac Delta Function: d dx STEP[x] = d dx  x −∞ δ[α] dα = δ[x] − δ[−∞] = δ[x] − 0 = δ[x] (6.103a) Therefore, the derivative of a step function centered at x 0 is a Dirac delta function located at x 0 : d dx STEP[x − x 0 ] = δ[x − x 0 ] (6.103b) Another and very important representation of The Dirac Delta Function is obtained by summing complex sinusoids with unit amplitudes and zero phase over all spatial frequencies:  +∞ −∞ e +2πiξx dξ =  +∞ −∞ cos[2πξx] dξ + i  +∞ −∞ sin[2πξx] dξ (6.104) To see that the expression satisfies the criteria required of The Dirac Delta Function in Equation (6.88), the integral may be evaluated over arbitrary finite and symmetric limits:  +B −B e +2πiξx dξ = 1 2πix (e +2πξx )| ξ =B ξ =−B = 1 πx (e +2πixB − e −2πixB ) 2i = 1 πx sin[2πBx] = 2B SINC[2Bx] (6.105) 1-D Special Functions 131 In the limit B → +∞, this expression is equivalent to Equation (6.96a) and thus is a valid representation of The Dirac Delta Function. Note that the integral of the original complex Hermitian function over symmetric limits yields a real-valued result due to the cancellation of areas for positive and negative x in the antisymmetric imaginary part. The representations of δ[x] as a summation of unit-amplitude cosines over all spatial frequencies may be visualized by considering the partial sum of sinusoidal components illustrated in Figure 6.25. Four cosine functions with periods of 4, 6, 8, and ∞ are added; obviously, the last cosine with infinite period is identical to the unit constant.

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  Mathematical Foundations of Imaging, Tomography and Wavefield Inversion

  • Anthony J. Devaney(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  4 Radiation and initial-value problems for the wave equation φ(0) =  ∞ −∞ dt δ(t)φ(t), χ (0) =  d 3 r δ(r)χ (r), (1.10) where φ(t) and χ (r) are any well-behaved ordinary functions of t and r, respectively. The Dirac Delta Functions do not have meaning within the framework of classical function the- ory and must be interpreted within the framework of distribution theory, where the “inte- grals” in the above definitions are taken to be inner products defined on a suitable space of “testing functions.” Although δ(t) and δ(r) are not ordinary functions, they can be formally manipulated and treated as such as long as at the end of a calculation they appear in inte- grals with ordinary functions that can then be given meaning through Eqs. (1.10). In this connection, we note that the Fourier transforms of the delta functions are given by 1 =  ∞ −∞ dt δ(t)e iωt , 1 =  d 3 r δ(r)e −iK·r , (1.11a) which follows from Eqs. (1.10) on taking φ(t) = exp(iωt) and χ (r) = exp(−iK · r). The delta functions then admit Fourier-integral representations given by δ(t) = 1 2π  ∞ −∞ dω e −iωt , δ(r) = 1 (2π ) 3  d 3 K e iK·r . (1.11b) We will interpret the Fourier integral throughout this book within the context of distri- bution theory, which amounts to using the transforms and inverse transforms in a purely formal way without any regard for the properties of the functions being transformed or inverse transformed. In most cases the results we obtain will hold within the classical the- ory of the transform but will be obtained using much less effort than would be required using the classical theory. In some cases the results cannot be obtained using classical the- ory but have a perfectly acceptable interpretation within distribution theory as, for example, will be the case when we compute the Green function of the wave equation in the follow- ing section. We will not detour into a review of distribution theory but will present certain results from the theory when needed.

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